Object variables, array

Figure 2 shows the code for the spin-echo pulse sequence. In lines 1-7, NMR object variables are declared. In lines 9-16, these objects are assigned values or specific NMR operations. A lactate spin system is read in from a file using a standard GAMMA text file format. Lines 20-24 are the code for the actual 90°Y-delay-180°Y-delay-acquire pulse sequence. In line 26, the density matrix is parsed into a transition table for the specified observation operator. And the "write results" function in line 27 converts the GAMMA transition table into three arrays of ppm, area and phase values one value for each line found in the transition table. This is a fairly trivial example, and the use of ideal pulses is often not sufficient to account for real-world artefacts in a simulation, but it shows how the object-oriented style of coding results in short amounts of code that is easy to read and comprehend. When compiled with the Visual Studio C++ compiler on a... 

Equating a variable in VBA to a worksheet range automatically creates an array in VBA in which each array element contains the value stored in the cell. Using the Set command to equate an object variable in VBA to a worksheet range automatically creates an array of "cell" objects in VBA. If you change the value of one of the array elements by means of your VBA code, the value in that cell in the worksheet will change, as illustrated by the Sub procedure in Figure 15-3. 

For an array of object variables, you must use a different approach to obtain the upper or lower bounds of the array indices, thus ... 

In the case of a cubic core array a series of orthogonal rotations can be performed on the three spaces of the objects, variables, and conditions, looking for the common orientation for which the core array is as much body-diagonal as possible. 

A data matrix can be considered as a two-way array, with the objects and variables forming the two different ways. In some applications it is necessary to extend this scheme to multiway arrays. For example, a three-way array (a rectangular block of data, Figure 3.21) occurs if the variables of objects are measured at various time points applications in chemistry are data from hyphenated methods (chromatography combined with spectroscopy). There are several possibilities to analyze such data only basic ideas are presented here. The interested reader is referred to Smilde et al. (2004). 

Multivariate Image Analysis Strong and Weak Multiway Methods Strong and weak -way methods analyze 3D and 2D matrices, respectively. Hyperspectral data cube structure is described using chemometric vocabulary [17]. A two-way matrix, such as a classical NIR spectroscopy data set, has two modes object (matrix lines) and V variables (matrix columns). Hyperspectral data cubes possess two object modes and one variable mode and can be written as an OOV data array because of their two spatial directions. 

The other instance variables in the Compound class include a Molstructure object, a list of salt structures, a list of parent structures, a String array that contains information about what has been changed or what the problems are in the structure as a result of the chemistry rules checking, and a Compounds tate object representing the current state of the Compound object. The process() method in the Compound object simply delegates the call to its current state object, which knows how to process the Compound object depending on its current state as described above. 

The main goal of this section is to provide a summary of several of the most widely used multivariate procedures in food authentication out of the vast array currently available. These are included in well-known computer packages such as BMDP, IMSL, MATLAB, NAG, SAS, SPSS and STATISTIC A. The first three subsections describe unsupervised procedures, also called exploratory data analysis, that can reveal hidden patterns in complex data by reducing data to more interpretable information, to emphasize the natural grouping in the data and show which variables most strongly influence these patterns. The fourth and fifth subsections are focused on the supervised procedures of discriminant analysis and regression. The former produces good information when applied under the strictness of certain tests, whereas the latter is mainly used when the objective is calibration. 

The data shown in Table 7 comprise a portion of a multi-element analysis of mineral water samples. The data from such an analysis can conveniently be arranged in an n by m array, where n is the number of objects, or samples, and m is the number of variables measured. This array is referred to as the data matrix and the purpose of using matrix notation is to allow us to handle arrays of data as single entities rather than having to specify each element in the array every time we perform an operation on the data set. Our data matrix can be denoted by the single symbol X and each element by Xy, with the subscripts i and j indicating the number of the row and column respectively. A matrix with only one row is termed a row vector, e.g., r, and with only one column, a column vector, e.g., c. 

Figure 1.1. A two-way array has, e.g., the objects as rows and the variables as columns. There are I objects and J variables.

Figure 1.2. A typical three-way array has I objects, J type 1 variables and K type 2 variables.

For a three-way array of liquid-chromatography-ultraviolet spectroscopy data, the samples are the objects whereas the absorbance at different retention times (of the chromatography mode) and the wavelengths (of the ultraviolet spectroscopy mode) are the two variable modes. Hence, absorbance is measured as a function of two properties retention time and wavelength. For a three-way array of food products x judges x sensory properties, the judge mode can be an object mode or a variable mode. In a preliminary analysis the judges... 

Some three-way models do not take the nature of a way into account, but the concepts of objects and variables are important, e.g., for defining how an array is preprocessed and for interpretation of the results. The same naturally holds for two-way analysis methods. 

In principal component analysis (PCA), a matrix is decomposed as a sum of vector products, as shown in Figure 1.6. The vertical vectors (following the object way) are called scores and the horizontal vectors (following the variable way) are called loadings. A similar decomposition is given for three-way arrays. Here, the array is decomposed as a sum of triple products of vectors as in Figure 1.7. This is the PARAFAC model. The vectors, of which there are three different types, are called loadings. 

Figure 1.6. In principal component analysis, a two-way array is decomposed into a sum of vector products. The vectors in the object direction are scores and the ones in the variable direction are loadings.

A two-way array or data matrix usually has rows representing objects and columns representing variables. This array has I x J entries, e.g. I objects and J variables measured on each object. 

There are different ways to derive the Tucker3 model. One way is nicely illustrated by Kroo-nenberg [1984], Assume that a three-way array X (I x / x K) is available with elements Xyk representing the measurement of the th variable on the zth object at the kth occasion. Suppose that two idealized objects are present, pi and p2 two latent variables are present, qi and q2 and two prototype occasions are present, ri and r2. The scores of all the idealized... 

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